CRITICAL TWO-POINT FUNCTIONS FOR LONG-RANGE STATISTICAL-MECHANICAL MODELS IN HIGH DIMENSIONS

We consider long-range self-avoiding walk, percolation and the Ising model on Z(d) that are defined by power-law decaying pair potentials of the form D(x) asymptotic to vertical bar x vertical bar(-d-alpha) with alpha > 0. The upper-critical dimension d(c) is 2(alpha boolean AND 2) for self-avoiding walk and the Ising model, and 3(alpha boolean AND 2) for percolation. Let alpha not equal 2 and assume certain heat-kernel bounds on the n-step distribution of the underlying random walk. We prove that, for d > d(c) (and the spread-out parameter sufficiently large), the critical two-point function G p(c) (X) for each model is asymptotically C vertical bar x vertical bar(alpha boolean AND 2-d), where the constant C is an element of (0, infinity) is expressed in terms of the model-dependent lace-expansion coefficients and exhibits crossover between alpha < 2 and alpha > 2. We also provide a class of random walks that satisfy those heat-kernel bounds.